Preheat the oven to 335F (170C or 160C in a fan forced oven).
Line the bottom of two 8-inch round cake pans with parchment paper. Grease and flour the sides. **Cake pans should have high sides (about 2 inches). See recipe notes for pan size alternatives.
Sift together the cake flour, baking powder, baking soda and salt in a medium bowl. Whisk a few times. Set aside.
Stir together the milk, oil and vanilla extract in a liquid measuring cup. Set aside.
In a very large bowl, beat together the butter and sugar until fluffy (about 3-5 minutes).
Beat in the egg whites into the butter a little at a time, turning off the mixer and scraping down the sides after each addition. I added mine in 3-4 additions.
With the mixer on a low speed, beat about ⅓ of the flour mixture into the butter mixture. Then mix in the milk mixture. Turn off the mixer and scrape down the sides of the bowl.
Beat in about ⅓ more of the flour mixture. I typically do a few stirs by hand first. Add in the champagne. First stir the batter a few times by hand, then mix with the electric mixer starting on a low speed.
Beat in the last ⅓ of the flour mixture. Once everything is combined stop mixing.
Pour the batter evenly in the prepared pans and smooth the tops.
Bake in the preheated oven with both cake pans on the same rack in the middle of the oven. Bake for 30-35 minutes, or until an inserted toothpick/cake tester comes out clean. You may start to see the cakes pulling away from the sides of the pan.
Cool cakes in the pans for at least 10-15 minutes. Then gently invert onto a cooling rack to continue cooling. To cool more quickly, after inverting onto the cooling rack, wrap them tightly in plastic and pop in the freezer. Make sure to wrap them in plastic to keep the moisture in.